Parallel Variable Distribution Algorithm for Constrained Optimization with Nonmonotone Technique

A modified parallel variable distribution (PVD) algorithm for solving large-scale constrained optimization problems is developed, which modifies quadratic subproblem 𝑄𝑃 𝑙 at each iteration instead of the 𝑄𝑃 0𝑙 of the SQP-type PVD algorithm proposed by C. A. Sagastiz´abal and M. V. Solodov in 2002. The algorithm can circumvent the difficulties associated with the possible inconsistency of 𝑄𝑃 0𝑙 subproblem of the original SQP method. Moreover, we introduce a nonmonotone technique instead of the penalty function to carry out the line search procedure with more flexibly. Under appropriate conditions, the global convergence of the method is established. In the final part, parallel numerical experiments are implemented on CUDA based on GPU (Graphics Processing unit).


Introduction
In this paper, we consider the following nonlinear programming problem: min  () , where  :   →  and () :   →   are all continuously differentiable.Suppose the feasible set  in (1) is described by a system of inequality constraints: In this paper, we give a new algorithm based on the method in [1], which partitions the problem variables  ∈ Parallel variable distribution (PVD) algorithm for solving optimization problems was first proposed by Ferris and Mangasarian [2], in 1994, in which the variables are distributed among  processors.Each processor has the primary responsibility for updating its block of variables while allowing the remaining secondary variables to change in a restricted fashion along some easily computable directions.The distinctive novel feature of this algorithm is the presence of the "forget-me-not" term which allows for a change in "secondary" variables.This makes PVD fundamentally different from the block Jacobi [3] and coordinate descent [4] methods.The forget-me-not approach improves robustness and accelerates convergence of the algorithm and is the key to its success.In 1997, Solodov [5] proposed useful generalizations that consist, for the general unconstrained case, of replacing exact global solution of the subproblems by a certain natural sufficient descent condition, and, for the convex case, of inexact subproblem solution in the PVD algorithm.Solodov [6] proposed an algorithm applied the PVD approach to problems with general convex constraints directly use projected gradient residual function as the direction and show that the algorithm converges, provided that certain conditions are imposed on the change of secondary variables.In 2002, Sagastizábal and Solodov [1] proposed two new variants of PVD for the constrained case.Without assuming convexity of constraints, but assuming blockseparable structure, they showed that PVD subproblems can be solved inexactly by solving their quadratic programming approximations.This extends PVD to nonconvex (separable) feasible sets and provided a constructive practical way of solving the parallel subproblems.For inseparable constraints, but assuming convexity, they developed a PVD method based on suitable approximate projected gradient directions.The approximation criterion was based on a certain error bound result, and it was readily implementable.In 2011, Zheng et al. [7] gave a parallel SSLE algorithm, in which the PVD subproblems are solved inexactly by serial sequential linear equations, for solving large-scale constrained optimization with block-separable structure.Without assuming the convexity of constraints, the algorithm is proved to be globally convergent to a KKT point.Han et al. [8] proposed an asynchronous PVT algorithm for solving large-scale linearly constrained convex minimization problems with the idea of [9] in 2009, which based on the idea that a constrained optimization problem is equivalent to a differentiable unconstrained optimization problem by introducing the Fischer Function.And in particular, different from [9] the linear rate of convergence does not depend on the number of processors.
In this paper, we use [1] as our main reference on SQPtype PVD method for problems (1) and (4).Firstly, we introduce the algorithm in [1] simply.The original problem is distributed into  parallel subproblems which treatment among  parallel processors.The algorithm of [1] may result in that the linear constraints in quadratic programming subproblems are inconsistent or the solution of quadratic programming subproblems is unbounded, so that the algorithm may fail.This drawback has been overcome by many researchers, such as [10].In [1], exact penalty function is used as merit function to carry out the line-search.For the penalty function, as pointed out by Fletcher and Leyffer [11], the biggest drawback is that the penalty parameter estimates could be problematic to obtain.To overcome this drawback, we use a nonmonotone technique to carry out the line search instead of penalty function.
Recent research [12][13][14] indicates that the monotone line search technique may have some drawbacks.In particular, enforcing monotonicity may considerably reduce the rate of convergence when the iteration is trapped near a narrow curved valley, which can result in very short steps or zigzagging.Therefore, it might be advantageous to allow the iterative sequence to occasionally generate points with nonmonotone objective values.Grippo et al. [12] generalized the Armijo rule and proposed a nonmonotone line search technique for Newton's method which permits increase in function value, while retaining global convergence of the minimization algorithm.In [15], Sun et al. give several nonmonotone line search techniques, such as nonmonotone Armijo rule, nonmonotone Wolfe rule, nonmonotone F-rule, and so on.Several numerical tests show that the nonmonotone line search technique for unconstrained optimization and constrained optimization is efficient and competitive [12][13][14]16].Recently, [17] gives a method to overcome the drawback of zigzagging with nonmonotone line search technique to determine the step length, which makes the algorithm more flexible.
In this paper, we combine [1] with the ideas of [10,17] and propose an infeasible SQP-type Parallel Variable Distribution algorithm for constrained optimization with nonmonotone technique.
The paper is organized as follows.The algorithm is presented in Section 2. In Section 3, under mild assumptions, some global convergence results are proved.The numerical results are shown in Section 4. And conclusions are given in the last section.
We use   ∈  to perturb the subproblem of  0  of [1] and give a new subproblem   instead of it.Consider For convenience, given , the th block of ∇() will be denoted by In (5),    is   in step .Note that ( 5) is always feasible for   = 0,   = max ∈  {  (   ), 0}.To test whether constraints are satisfied or not, we denote the violation function ℎ() as follows: where   () + = max{  (), 0},  ∈ , ‖ ⋅ ‖ denotes the Euclidean norm on   .It is easy to see that ℎ() = 0 if and only if  is a feasible point (ℎ() > 0 if and only if  is infeasible).We describe our modified PVD algorithm as follows.
Algorithm A. Consider the following.
Having   , check a stopping criterion.If it is not satisfied, compute  +1 as follows.
In a restoration algorithm, we aim to decrease the value of ℎ() more precisely, we will use a trust region type method to obtain ℎ(  ) → 0,  → ∞ by the help of the nonmonotone technique.Let and    =    ()/Ψ   ().Algorithm B is similar to the restoration phase given by Su and Yu [17], we describe the Restoration Algorithm as follows.
Algorithm B. Consider the following.

The Convergence Properties
To prove the global convergence of Algorithm A, we make the following assumptions.

Assumptions
where  2 is a constant.Remark 3. We can use quasi-Newton BFGS methods to update    , different from [18], we can use a small modification of    to make    reserve positive definite according to the formula (33) of [17].

Journal of Applied Mathematics
Similar to Lemma 1 in [19], we can get the following conclusions.
Proof.First note that the feasible set for (5) Since the function being minimized in ( 14) is strictly convex and radially unbounded, it follows that    is well defined and unique as a global minimizer for the convex problem (5) and thus unique as a KKT point for that problem.So we have due to (17)  (B1) The following inequality holds: where Φ  (  ) = max ∈  {  (  ), 0}.
From Lemmas 6 and 7, we know that the Algorithm A and Algorithm B are well implemented.
Proof.By Assumption (A1), there exists a point  * such that   →  * for  ∈ , where  is an infinite index set.By Algorithm A and Lemma 8, we consider the following two possible cases.

Now for any 𝑘, 𝑥
Since {( () )} admits a limit, by the uniform continuity of  on , it holds lim Then by the relation (29), we have lim Using the same arguments for deriving (33), we obtain that lim Together with (44), (47), and transpose Taking into account separability of constraints (), then which contradicts the definition of  1 .The proof is complete.Taking into account separability of constraints () and , we conclude Therefore  * is a KKT point of problem (1).

Numerical Results
To get some insight into computational properties of our approach in Section 2, we considered the same test problems taken from [1].Choose Problem 1 of [1] as follows: In our test, we only verify our convergence theory without comparing with serial algorithms.In the future, we will propose the convergent rate of the parallel algorithm.Not having access to a parallel computer, we have carried out on Graphics Processing Unit (GPU) to solve them and implement the algorithm on Compute Unified Device Architecture (CUDA) [20].However, the input and output are implemented by CPU.And all the subproblems are solved on blocks which are constructed into many threads.If there are  blocks in GPU, which is equivalent to  processors in a parallel computer.Many threads of the block can achieve a large number of vector calculations in current block.
We have implemented Algorithm A in C language.All codes were run under Visual Studio 2008 and Cuda 4.0 on the DELL Precision T7400.In Table 1, we report the number of iterations and the running times for Algorithms A on Problem (52).The results in Table 1 confirm that the proposed approach certainly makes sense.We solve the subproblem (5) which is a positive semidefinite quadratic programs by the modified interior-point methods.And the Restoration Algorithm of [17] is solved by Trust-region Approach combined with local pattern search methods.All algorithms are coded by C language without using any function from optimization Toolbox; hence the results are not the best.
In Table 1,  denotes the dimensions of test problem and ] denotes the variable number of one block.There is less iteration number of   subproblem with more numbers of blocks for high-dimensional problem, which shows the algorithm is reliable.

Conclusion
The PVD algorithm which was proposed in 1994 is used to solve unconstrained optimization problems or has a special case of convex block-separable constraints.In 1997, M. V. Solodov proposed useful generalizations that consist, for the general unconstrained case, of replacing exact global solution of the subproblems by a certain natural sufficient descent condition, and, for the convex case, of inexact subproblem solution in the PVD algorithm.M. V. Solodov proposed a PVD approach in 1998 which applied to problems with general convex constraints directly and show that the algorithm converges.In 2002, C. A. Sagastizábal et al. propose two variants of PVD for the constrained case: without assuming convexity of constraints, but assuming block-separable structure and for inseparable constraints, but assuming convexity.In this paper, we propose the modified algorithm of [1] used to the general constraints but with block-separable structure.
In a word, the algorithm above is a special structure of the objective function or constraints with a special structure.In the further, we will study the parallel algorithm with general inseparable constraints.

Remark 2 .
Assumptions (A1) and (A2) are the standard assumptions.(A3) is the LICQ constraint qualification.(A4) plays an important role in obtaining the convergence results.(A5) is the sufficient reduction condition which guarantees the global convergence in a trust region method.Under the assumptions,  is bounded below and the gradient function   ,  = 1, . . .,  is uniformly continuous in   , where  = ( 1 , . . .,   ).

Table 1 :
Results for Algorithm A.